Substrate constraint modifies the Rayleigh spectrum of vibrating sessile drops

Abstract

In this work, we study the resonance behavior of mechanically oscillated, sessile water drops. By mechanically oscillating sessile drops vertically and within prescribed ranges of frequencies and amplitudes, a rich collection of resonance modes are observed and their dynamics subsequently investigated. We first present our method of identifying each mode uniquely, through association with spherical harmonics and according to their geometric patterns. Next, we compare our measured resonance frequencies of drops to theoretical predictions using both the classical theory of Lord Rayleigh and Lamb for free, oscillating drops, and a prediction by Bostwick and Steen that explicitly considers the effect of the solid substrate on drop dynamics. Finally, we report observations and analysis of drop mode mixing, or the simultaneous coexistence of multiple mode shapes within the resonating sessile drop driven by one sinusoidal signal of a single frequency. The dynamic response of a deformable liquid drop constrained by the substrate it is in contact with is of interest in a number of applications, such as drop atomization and ink jet printing, switchable electronically controlled capillary adhesion, optical microlens devices, as well as digital microfluidic applications where control of droplet motion is induced by means of a harmonically driven substrate.

Footprint geometry and sessile drop resonance

“…A final remark on dispersion: figure 8 is consistent with our prior work [21] for zonal and sectoral modes. Namely, the resonance frequency does not scale with wavenumber as RayleighLamb theory would predict (f ∼k 1.5 ) [38,39].…”

“…Square and circular drops are tested with the same approach as in our previous works [21,22]. can be clearly resolved on the platform.…”

“…The free surface waves oscillate at the forcing frequency (harmonically) when the forcing acceleration is low. However, an elevated acceleration triggers modes which oscillate at half the forcing frequency [15,18,19,21,[30][31][32], called half-frequency subharmonic modes.…”

“…(For a planar rest state, the Laplace-Beltrami operator reduces to the classical Laplace operator.) This model has proved successful in capturing resonant frequencies for driven sessile drops that are not too flat [12,21,22]. These standing waves, pinned along circular footprints, are therefore classified using the spherical harmonic Square Drop Resonance February 21, 2017 Chang, Daniel, Steen symmetries inherited from free spherical drop oscillations [38,39].…”

“…For both, typical observables are mode shapes [11,[14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32], resonance frequencies [11,[14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32], and the evolution of surface waves with forcing amplitude [18,23,[29][30][31][33][34][35]. The free surface waves oscillate at the forcing frequency (harmonically) when the forcing acceleration is low.…”